Introduction:
In mathematics, quadratic equations are the polynomial equation of the second degree. The general form is,
ax^2 + bx + c = 0
where x represents the variable, and a, b, and c, constants, with a ? 0. (If a = 0, the equation becomes a linear equation.)
The constants a, b, and c respectively known as the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" came from quadratus, which is the Latin word for "square." Quadratic equations could be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below). One common use of quadratic equations are to compute trajectories in projectile motion.
Quadratic Formula Explanation:
Often, the simplest way to solve "ax^2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. While factoring could not always be successful, the Quadratic Formula can always find the solution.
The Quadratic Formula uses the co-efficient "a", "b", and "c" from "ax^2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients". The Formula is derived from the process to complere the square, and is formally stated as:
For ax^2 + bx + c = 0, the value of x is given by: x = (-b ± v(b2 - 4ac)) / 2a
Understanding Roots of a Quadratic Equation is always challenging for me but thanks to all math help websites to help me out.
Discriminant:
In the above formulae, the expression underneath the square root sign is called the discriminate of the quadratic equation, and is often represented using an upper case Greek Delta:
? = b^2 - 4ac
A quadratic equation with real co-efficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant defines the number and nature of the roots. There are three cases:
(1) If the discriminant is positive, then there should be two distinct roots, both of which are real numbers. For quadratic equations with integer co-efficients, if the discriminant is a perfect square, then the roots are rational numbers in other cases they may be quadratic irrationals.
(2) If the discriminant is zero, then there must be exactly one distinct real root, sometimes called a double root:
x = -b/2a.
(3) If the discriminant is negative, then there is no real roots. Rather, there are two distinct complex roots, which are complex conjugates of each other.
Thus the roots may distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
In mathematics, quadratic equations are the polynomial equation of the second degree. The general form is,
ax^2 + bx + c = 0
where x represents the variable, and a, b, and c, constants, with a ? 0. (If a = 0, the equation becomes a linear equation.)
The constants a, b, and c respectively known as the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" came from quadratus, which is the Latin word for "square." Quadratic equations could be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below). One common use of quadratic equations are to compute trajectories in projectile motion.
Quadratic Formula Explanation:
Often, the simplest way to solve "ax^2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. While factoring could not always be successful, the Quadratic Formula can always find the solution.
The Quadratic Formula uses the co-efficient "a", "b", and "c" from "ax^2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients". The Formula is derived from the process to complere the square, and is formally stated as:
For ax^2 + bx + c = 0, the value of x is given by: x = (-b ± v(b2 - 4ac)) / 2a
Understanding Roots of a Quadratic Equation is always challenging for me but thanks to all math help websites to help me out.
Discriminant:
In the above formulae, the expression underneath the square root sign is called the discriminate of the quadratic equation, and is often represented using an upper case Greek Delta:
? = b^2 - 4ac
A quadratic equation with real co-efficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant defines the number and nature of the roots. There are three cases:
(1) If the discriminant is positive, then there should be two distinct roots, both of which are real numbers. For quadratic equations with integer co-efficients, if the discriminant is a perfect square, then the roots are rational numbers in other cases they may be quadratic irrationals.
(2) If the discriminant is zero, then there must be exactly one distinct real root, sometimes called a double root:
x = -b/2a.
(3) If the discriminant is negative, then there is no real roots. Rather, there are two distinct complex roots, which are complex conjugates of each other.
Thus the roots may distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
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